Tuesday, June 16, 2015

Mathiness is next to growthiness (the 4% solution)

4% average growth has not happened since the 1970s.

I stole Sandwichman's excellent title from EconoSpeak.

John Cochrane put up a graph today trying to persuade us that Jeb Bush's goal of 4% real GDP growth is possible contra several other economists. I tried to figure out what he did because it couldn't have been a simple moving average (orange curve). His graph went until 2015 so it would have to be a backward looking average of some kind. But then both a backward looking moving average (green dashed) and a backward looking integral average (blue, they're basically on top of each other) don't quite make it up to 4% since the 1970s.

I did eventually figure it out (the red curve), but it involves the same mistake that I took Scott Sumner to task for back awhile ago. Cochrane took the change in RGDP from a point t - 10 to t and divided by RGDP at t. That gives the slope of the secant from the start point to the end point.

It may be called the mean value theorem, but that doesn't mean that the secant gives an average value. All it does is say there is at least one point that has that slope -- that 4% growth happened sometime during that 10 year period.

These diagrams might make this clearer. The first one (on the left) shows an RGDP function that gives 4% growth between the endpoints 10 years apart. You can see there is a point just after year 8 that has a tangent with the same slope. In the second graph (on the right) you can see that this function has nearly zero growth across the entire 10 year period. Not many people would consider that averaging 4% RGDP growth (it actually averages around 3.5% growth during that entire period per the averaging methods above). Cochrane's method over-weights the high growth spikes.

The method other economists are using to say growth has been less than 4% is the better method. Of course, you can choose to use the mean value theorem method if you want to overweight growth spikes to make growth to look higher ... and you're into mathiness.


  1. Maybe I'm just crazy but I would have thought that most economists would see 4% growth as not feasible for reasons other than the low growth of the 2010's... Maybe they think that participation rates in the labor market will return to normal or something. As I understand it, though, the ITM projects NGDP (and therefore RGDP) to slow down as the economy gets large, but I'm not sure a sudden drop from 4% trend to 2% trend really captures the essence of a sudden increase in the probability that money will be used in a slow growth market. Does the ITM give any reasons for an almost instantaneous halving of RGDP growth rates?

    1. Hi John,

      The graphs at the top are just moving averages -- in the model, RGDP growth slows down smoothly over time.

      See e.g. Here:


    2. So, if I'm interpreting correctly, the ITM explains the lull in (moving average) RGDP growth as partially that gradual slow down and partially a random walk.

    3. Sort of, but there is more to it. The negative deviations may be a result of non-ideal information transfer as well as random walk. So it is:

      trend + non-ideal + random component (with unit root)

      ... however there is no model as yet for the depth of the non-ideal deviations.

  2. Sadly you and Cochrane both fall to Krugman 's level of earning incompletes for your econometrics homework. I'm sure if you complain to the prof, she'll grade it herself, she is always more lenient than me.... Cochrane for using an averaging method that trolls love to troll and you for arguing why someone else is wrong without explaining for instance why sensitivity to high growth spikes are more important than its sensitivity to deep falls. Instead you should have written an article supporting the thesis that say an ma is better or at least demonstrated how it is immune to these problems...no points for mathiness all around

    1. (Also note I'm using a classic lecture session trap...I really want you to say that sensitivity to high growth peaks is more important because more of them happen...)

    2. Hi LAL,

      The moving averaging methods (integral and simple average) are less sensitive to spikes in either direction -- the spikes have vanishing support in the limit of your measurement delta-t going to zero so they contribute zero to the integral.

      In the mean value theorem method, every spike contributes (in either direction). Imagine a step function RGDP: it would have zero growth in most time periods but with a spike of almost infinite growth in the middle resulting in an "average" growth that was entirely dependent on the size of one spike.

      However this is a good audition for you to take over as the grumpy economist when Cochrane gives up blogging :)

    3. I don't think I fell in your trap :)

    4. I guess in the context of the articles this would have gotten an A, in the class room I would still act dumbfounded as to why throwing away business cycles is important to my students...why shouldn't we weight upswings more than no movement? That's the economic question....and why we are in econometrics and not time series analysis...Also that is the nicest thing anyone has ever said to me :D

  3. Actually I'm having trouble matching Cochrane description to yours...

    1. I think there were some small differences due to his use of annual data where I used quarterly but it matched up with his original graph.

      However, it appears as though he's updated his graph to include something more like the average I did.

    2. His average seems to be (present -past)/past...his formula is the closest I had to matching his picture but ....still I'm not matching...I would use log(st/st-1)...Also I'm pretty sure he was straight up trolling me UN the comments section

    3. apparently the UN has earned enough ire from me to replace the word "in" as default on my phone...

  4. I somehow managed to fail to copy and paste...B effort...I am now matching...

  5. Hmmm. I would not even start with a graph of growth rates. I would use log(1+r). A decade of 4% yearly increases produces an increase of 48%, not 40%. If you don't use logs, then what you want for a simple moving average is (RGDP(t+5)/RGDP(t-5))^(0.1). That's my mathiness. :)

    1. For the exponentially challenged (those who have trouble with tenth roots), here is a tip.

      It seems that Cochrane used the following formula:

      mean_growth_rate = ((RGDP(t = 0) - RGDP(t = -10))/RGDP(t = -10))/10

      = (RGDP(t = 0) - RGDP(t = -10))/10*RGDP(t = -10)

      One peculiar thing about that equation that stands out is that it gives added weight to RGDP(t = -10) for no apparent reason. If we apply a little common sense we can weight the two RGDP values equally and get this.

      mean_growth_rate = (RGDP(t = 0) - RGDP(t = -10))/(5*RGDP(t = -10) + 5*RGDP(t = 0))

      Let's try that with a 4% annual growth rate. Let RGDP(t = -10) = 1 and RGDP(t = 0) = 1.48. Then we have

      mean_growth_rate = (1.48 - 1)/(5 + 7.4) = 0.48/12.4 = 0.0387

      A slight under-estimate, but good enough for government work. And well within the error of economic calculations. :)


      Cochrane is aware of using logarithms but pooh-poohs doing so, choosing instead to use a calculation that obviously overestimates the annual growth rate. {sigh}

    2. Sorry. I am having fun. :)

      The harmonic mean is often preferable to the arithmetic mean, and it looks like that would be the case here, as well. So if we divide by the harmonic mean of RGDP(t = 0) and RGDP(t = -10) we get his equation.

      mean_growth_rate = (RGDP(t = 0)^2 - RGDP(t = -10)^2)/20*RGDP(t = -10)*RGDP(t = 0)

      With the 4% example above that gives us

      mean_growth_rate = 1.19/29.6 = 0.0402

      Not bad. :)

  6. As for Jeb Bush aiming at 4% growth in RGDP, maybe he is taking advice from his father. Something like this:

    Bush pere: Learn from your brother. I always said that this Reaganomics supply side shit was voodoo economics. Like Nixon said, we're all Keynesians now.


    No disparagement of the information transfer approach. IMO the Reagan revolution really was a persistent non-random phenomenon. Pre- and post-1980 really are different politico-economic regimes.

  7. You guys probably already noticed but John Cochrane. did provide an update with the exact .m code he used to do the calculation.

    O/T: Jason, you probably already saw this as well, but Scott writes in his latest:

    "While I’m impressed by an explanation that’s as flexible as a circus contortionist, I’d prefer something that isn’t consistent with any possible state of the universe. I’m no Popperian, but I like my theories to be at least a little bit falsifiable."

    1. I did see it, thanks Tom.

      It's flabbergasting but then again as humans we frequently see faults in others that we fail to see in ourselves.

      So does this mean Scott knows a way market monetarism can be falsified?

    2. “O, wad some Power the giftie gie us
      To see oursels as others see us!
      It wad frae monie a blunder free us,
      An' foolish notion.”

      -- Robert Burns

    3. "So does this mean Scott knows a way market monetarism can be falsified?"

      I asked him. Unfortunately I packed too many questions in my comment and he didn't answer that one. I had a feeling that might happen. I asked him for a concrete example or two of something that would falsify his theories at least a "little bit." (c:


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